A Lefschetz fixed point formula for elliptic quasicomplexes
Consider a sequence \((V^{\bullet}, A)\) of bounded linear operators \(A^i\in\mathcal {L}(V^i,V^{i+1})\) between Hilbert spaces, \(i=0,\dots, N-1\). Let \(\mathcal I\) be an operator ideal in the class of compact operators. The sequence \((V^{\bullet}, A)\) is said to be an \(\mathcal I\)-quasicomplex if \(A^{i+1}A^i\in\mathcal{I}(V^i,V^{i+2})\) holds for \(i=0,\dots, N-2\). An \(\mathcal{I}\)-quasiendomorphism of a quasicomplex \((V^{\bullet}, A)\) is a sequence \(E\) of linear operators \(E^i\in \mathcal {L}(V^i)\) such that \(E^{i+1}A^i-A^iE^i\in\mathcal{I}(V^i,V^{i+1})\), \(i=0,\dots,N-1.\) A quasicomplex is said to be Fredholm if it admits an \(\mathcal{I}\)-parametrix, that is, if there is a sequence of operators \(P^i\in\mathcal {L}(V^i,V^{i-1})\) satisfying \(P^{i+1}A^i+A^{i-1}P^i=\mathrm{Id}_{V^i}-R^i,\) with \(R^i\in \mathcal{I}(V^i),\) \(i=0,\dots, N.\) In a previous work [Math. Sci., Springer 6, Article ID 44, 8 p. (2012; Zbl 1301.47017)] \textit{N. Tarkhanov} and \textit{D. Wallenta} proved a result which implies that for any Fredholm \(\mathcal{I}\)-quasicomplex \((V^{\bullet}, A)\) there exist operators \(D^i\in\mathcal L(V^i,V^{i+1})\) such that \(D^i-A^i\in\mathcal{I}(V^i,V^{i+1})\) and \(D^{i+1}D^i=0.\) In this work the author proves that if \((V^{\bullet}, A)\) is a Fredholm \(\mathcal{I}\)-quasicomplex, \(E\) an \(\mathcal{I}\)-quasiendomorphism of this quasicomplex and \((V^{\bullet}, D)\) is any complex with \(D^i-A^i\in\mathcal{I}(V^i,V^{i+1})\) then there exists an endomorphism \(\tilde{E}\) of \((V^{\bullet}, D)\) such that \(\tilde{E}^i-E^i\in\mathcal{I}(V^i).\) If \(\mathcal{I}=\mathfrak{S}_1\) is the Schatten class (trace class operators), then \[ L(E,A)=\sum_{i=0}^{N}(-1)^i {\mathrm{tr }\tilde{E}^i+\sum_{i=0}^{N}(-1)^i \mathrm{tr }(E^i-\tilde{E}^i)} \] is well defined and it extends the classical Lefschetz number. As an application the author shows that in the case of an elliptic quasicomplex of pseudodifferential operators over a closed compact manifold \(X\) and a quasiendomorphism associated to a smooth selfmap of \(X\) having only simple fixed points the formula above generalizes the Lefschetz fixed point formula as in the work by \textit{M. F. Atiyah} and \textit{R. Bott} [Ann. Math. (2) 86, 374--407 (1967; Zbl 0161.43201)].
scientific article; zbMATH DE number 6351695
scientific article